Motion Planning

1. Motion Planning MajdSrour

2. Definition • Motion planning refers to the computational process of moving from one place to another in the presence of obstacles.

3. Work space • The physical space in which the robot of finite-size. – Real world

4. Configuration Space (Only translation) • The configuration space is a transformation from the physical space in which the robot is of finite-size into another space in which the robot is treated as a point. • The configuration space is obtained by shrinking the robot to a point, while growing the obstacles by the size of the robot.

5. Work space vs Configuration space

6. Free Space • The free space of a configuration space simply consists of the areas not occupied by obstacles.  Any configuration within this space is called a free configuration.

7. Free Path • The free path between an initial configuration and a goal configuration is the path which lies completely in free space and does not come into contact with any obstacles.

8. Motion Planning • Problems (from easy to hard) : • Completely knownstatic Obstacles • Completely known dynamic Obstacles • Partially Known static Obstacles • Partially Known dynamic Obstacles

9. Roadmap Approach • Visibility Graph • Voronoi method • Cell Decomposition

10. Visibility Graph • A visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane

11. Visibility Graph • Given a set S of disjoint (open) polygons..

12. Visibility Graph • Let V(S) be the vertex of S

13. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S.

14. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

15. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

16. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

17. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

18. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

19. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

20. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

21. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

22. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

23. Visibility Graph • Let V(S) be the vertex of S • Letbe the visibility graph of S. • Where:

24. Visibility Graph • Corollary: The shortest path between P_start and P_goal corresponds to a shortest path in

25. Finding the shortest path • We’ll use Dijkstra algorithm in order to find the shortest path in G • For each • return

26. Dijkstra • Assign to every node a distance value of infinity, and to the first node 0 • Mark all nodes as unvisited, set initial node as current • For current node, consider all of its unvisited neighbors and calculate their tentative distances • When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again; its distance recorded now is final and minimal. • If the destination node has been marked visited then stop. • Set the unvisited node marked with the smallest tentative distance as the next "current node" and go back to step 3.

27. Example • Find shortest path from a to z • http://www.youtube.com/watch?v=UG7VmPWkJmA